Let G be a simple graph with n vertices. Show that a) G is a tree if

Question:

Let G be a simple graph with n vertices. Show that
a) G is a tree if and only if it is connected and has n - 1 edges.
b) G is a tree if and only if G has no simple circuits and has n − 1 edges.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Discrete Mathematics and Its Applications

ISBN: 978-0073383095

7th edition

Authors: Kenneth H. Rosen

See More Books

Question Posted: May 03, 2016 01:57 AM

See More Questions

Let G be a simple graph. Show that the relation R on the set of vertices of G such that uRv if and only if there is an edge associated to {u, v} is a symmetric, ir-reflexive relation on G.
Show that if G is a simple graph with n vertices, then the union of G and is Kn.
Give a big-O estimate of the number of operations (comparisons and additions) used by Floyd's algorithm to determine the shortest distance between every pair of vertices in a weighted simple graph...
In Problems 530, a. Classify the sequences as arithmetic, geometric, Fibonacci, or none of these. b. If arithmetic, give d; if geometric, give r; if Fibonacci, give the first two terms; and if none...
List and describe the options available for the location of the information security functions within the organization. Discuss the advantages and disadvantages of each option.
Understand how the generation into which a consumer is born can influence their lifestyle choices and buying behaviours.
If you were a consumer who had installed Lumber Liquidators laminate flooring in your home or business, what would you do now?
The Voiture Company manufactures compact, energy efficient cars. On April 1, it purchased a machine for its assembly line at a contract price of $200,000 with terms of 2/10, n/30. The company paid...
Question 2 > Valerie recieves an invoice from Home and Office Supply dated May 3rd, 2020 for $1,250 with terms 6/10, 1/20, n/30. If Valerie pays the invoice in full on May 21st, 2020, a. How many...
5.18 Length of a movie on Netflix. Flixable reports that Netflix's U.S. catalog contains almost 4000 movies.2 You are interested in determining the average length of these movies. Previous studies...
a) How many nonisomorphic unrooted trees are there with five vertices? b) How many nonisomorphic rooted trees are there with five vertices (using isomorphism for directed graphs)?
How many edges does a full binary tree with 1000 internal vertices have?
Which of the following is not one of the categories of employee fraud symptoms? a. Accounting anomalies. b. Analytical anomalies. c. Tips and complaints. d. Firm structure.
Implement the nearest neighbor algorithm in the programming language of your choice. The algorithm should work with vectors of up to 10 integer values and allow up to 10 integer classifications. By...
Use the operators described in Section 16.2.4 and the STRIPS method to solve the block world planning problem shown in Figure 16.11. The first state shown is the start state and the second state is...
Implement a Bayesian belief network in the programming language of your choice to represent a subject in which you are interested (for example, you might use it to diagnose medical conditions from...
Researchers have measured the acceleration of racing greyhounds as a function of their speed; a simplified version of their results is shown in Figure P4.67. The acceleration at low speeds is...
If the rate at which energy is dissipated by resistor 1 in Figure P31.86 is $2.5 \mathrm{~W}$, and $R_{1}=10 \Omega, \mathscr{E}_{1}=12 \mathrm{~V}$, and $\mathscr{E}_{2}=6 \mathrm{~V},$ (a)...
Solve the systems in Problems 2130 graphically and indicate whether each solution region is bounded or unbounded. Find the coordinates of each corner point. 2x + y 2 16 = x + y = 12 x + 2y = 14. x 0...
Chicago Company sold merchandise to a customer for $1,500 cash in a state with a 6% sales tax rate. The total amount of cash collected from the customer was $558. $600. $642. $636. Nevada Company...
Sketch the graph of a function whose first and second derivatives are always negative.
A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what...
(a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. 11. f(x) = x' - 12.x...
September 1 . Purchased a new truck for $ 8 3 , 0 0 0 , paying cash. September 4 . Sold the truck purchased January 9 , Year 2 , for $ 5 3 , 6 0 0 . ( Record depreciation to date for Year 3 for the...
Find the NPV for the following project if the firm's WACC is 8%. Make sure to include the negative in your answer if you calculate a negative. it DOES matter for NPV answers
What is the value of a 10-year, $1,000 par value bond with a 12% annual coupon if its required return is 11%?

Let G be a simple graph with n vertices. Show that a) G is a tree if

Question:

Step by Step Answer:

a We will prove this statement using mathematical induction on n the number of vertices of G This exercise can also be done by using Exercise 14 and T...View the full answer

Discrete Mathematics and Its Applications

Students also viewed these Statistics questions